136

GENERAL EXERCISES.

(30) Show that the coefficient of expansion of a body may be found as follows:

:

Let s be the S.G. of the body at zero temperature compared with water at its greatest density; 1+e1, 1+e, the volumes at temperatures t1, t2 of a unit volume at zero temperature; 1+E1, 1+E2 the volumes at t1, t2 of a unit volume of water at its greatest density; w the weight of the body in a vacuum; w1, w, its apparent weights in water at temperatures t1, to; then

2

¤1 — е2 — E1 — E2 —s(w1—w1⁄2)/w very nearly.

=

(31) Prove that, if a hydrometer of weight W sinks to certain marks on the stem in a liquid at temperatures t1 and t2, and to the same marks in the liquid at zero temperature, when weights w, and w2 are fixed at the top of the hydrometer, the coefficients of cubical expression of the hydrometer and of the liquid are respectively

(32) Determine the s. v. in cubic feet to the ton, and the density in lb per cubic foot of lead shot, cast iron spherical shot, and cast iron spherical shells with internal radius three-quarters the outside radius, given the S.G. of lead as 114, and of cast iron 7.2.

Determine also the s.v. or roomage of earthenware pipes, and cylindrical barrels, of apparent density p.

CHAPTER IV.

THE EQUILIBRIUM AND STABILITY OF A SHIP OR FLOATING BODY.

84. Simple Buoyancy.

The Principle of Archimedes leads immediately, as in § 48, to the Conditions of Equilibrium of a body supported freely in fluid, like a fish in water, or a balloon in air, or like a ship floating partly immersed in water (fig. 38, p. 148). The body is in equilibrium under two forces;

(i.) its weight W acting vertically downwards through G, the C.G. of the body; and

(ii.) the buoyancy of the fluid, equal to the weight of the displaced fluid, and acting vertically upwards through B, the C.G. of the displaced fluid;

and for equilibrium these two forces must be equal and directly opposed.

The Conditions of Equilibrium of a body, floating like a ship on the surface of a liquid, are therefore

(i.) the weight of the body must be less than the weight of the total volume of liquid it can displace, or else the body will sink to the bottom of the liquid;

(ii) the weight of liquid which the body displaces in the position of equilibrium is equal to the weight W of the body;

(iii) the c.G. B of the displaced liquid and G of the body must lie in the same vertical line GB.

138

SIMPLE BUOYANCY.

85. In a ship the draft of water is a measure of the displacement and buoyancy of the water, while the freeboard, or height of the deck above the water line, is a similar measure of the reserve of buoyancy, or of the extra cargo which the ship can carry without sinking.

The Plimsoll mark is now, by Act of Parliament, painted on all British ships; it is a mark which must not be submerged when the vessel is floating in a fresh water dock, before putting to sea; and the mark is fixed at such a height as to give the vessel a reserve of buoyancy of 25 per cent. of its total buoyancy.

The buoyancy of a pontoon or cask, employed as a support or buoy, is however generally used to mean its reserve of buoyancy, or the additional weight required to submerge it.

Thus the (reserve of) buoyancy of a body, a life buoy for instance, of weight W lb and (apparent) S.G. s, and therefore displacing W/s lb of water, is

(-1) W lb.

86. When a ship loses its reserve of buoyancy, and is sunk in shallow water, it can be raised by building a caisson on the deck so as to bring the level of the bulwarks above the surface at low water.

All leaks and orifices below water having been stopped by divers, the vessel is pumped out at low water by powerful steam pumps; and thereby soon acquires sufficient buoyancy to rise from the bottom of the sea, so as to be moved into a dock for repair; in this manner such large vessels as the Utopia, the Austral, and the Howe have been raised.

THE CAMEL AND FLOATING DOCK.

139

When a vessel draws too much water for entering or leaving a port, as for instance Venice, the Zuyder Zee, or Chicago through the lakes of N. America, camels are employed to lessen the draft of water.

These camels consist of large tanks, which are submerged by the admission of water, and then secured to the sides of the vessel by chains passing under the keel. On being pumped out the extra buoyancy of the camels raises the vessel and lessens the draft of water to the desired extent.

The same principle is employed in floating docks: the dock is submerged by the admission of water, so that the vessel can be floated on to the blocks on the bottom of the dock and be there secured: the water is then pumped out of the dock and the vessel is thereby raised above the level of the water, and can then be deposited on staging ashore, or even repaired on the floating dock itself; in this case it is convenient to secure the dock to the quay wall by pivoted bars.

The double power dock, designed by Messrs. Clark and Stansfield, consists of a central pontoon which supports the vessel, and two large side tanks or camels, which can float independently. The vessel is raised as far as possible by pumping out the central pontoon; the camels are then submerged by the admission of water, and secured to the sides of the pontoon; and now, the buoyancy of these camels, on being pumped out, is sufficient to raise the vessel completely above the water.

By this arrangement not only is economy of material secured, but the pontoon or the camel can be alternately raised completely out of the water for the purpose of examination and repair. (Trans. I. Naval Architects, XX.)

140

TONS PER INCH IMMERSION.

87. Denoting by A the water line area (flottaison) of a ship in square feet, that is, the area of the plane curve formed by the water line, then an additional load of P tons properly placed (that is, so that the C.G. of P is vertically over or under the C.G. of the water line area) will cause the ship to draw h feet more water, of density D lb/ft3 suppose, given by the equation

Strictly speaking this supposes either that the ship is wall-sided, meaning that the sides of the ship in the neighbourhood of the water line form part of a cylindrical surface; or else that the mean water line area at the mean draft is A ft2; and thus, given P/h, we can determine A, and vice versa.

For sea water we take D=64, so that the s.v. of sea water is 2240÷64-35 ft3/ton; and

and P/h is the number of tons required to immerse the ship one inch.

Thus in a ship loading 10 tons to the inch, the water line area is 4200 ft2; and loading or consuming 300 tons of coal will change the draught 2 ft 6 in.

For a ship L ft long and B ft broad at the water line,

A=cLB,

where c is called the coefficient of fineness of the area. The following rules are given by Mr. W. H. White for a the coefficient of fineness, and n=P/h the number of tons per inch immersion (Naval Architecture) :—